Which number should replace ‘?’ in the following series? 240, 120, ?, 180,…
2023
Which number should replace ‘?’ in the following series?
240, 120, ?, 180, 360, 900
Answer: A. 120 — Concept — In a multiplicative series each term is produced by multiplying the previous term by a factor. When those factors themselves form an arithmetic…
- A.
120
- B.
130
- C.
150
- D.
More than one of the above
- E.
None of the above
Attempted by 28 students.
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Correct answer: A
Concept — In a multiplicative series each term is produced by multiplying the previous term by a factor. When those factors themselves form an arithmetic progression — that is, each factor is larger than the one before it by a constant step — the series is identified by extracting the factor of every consecutive pair that is known and checking whether those factors rise by a fixed amount. The missing term is then recovered from the factor that belongs in its position.
Application — Take the ratio of each consecutive pair whose two terms are both given:
240 to 120 gives a factor of 120 ÷ 240 = 0.5
180 to 360 gives a factor of 360 ÷ 180 = 2
360 to 900 gives a factor of 900 ÷ 360 = 2.5
There are six terms, so five factors link them. Reading them in order, the first factor is 0.5 and the last two are 2 and 2.5, giving the pattern 0.5, ?, ?, 2, 2.5. The known factors rise in steps of 0.5, so the two unknown factors are 1 and 1.5, and rebuilding the series factor by factor gives:
Factor 0.5: 240 × 0.5 = 120, the given second term.
Factor 1: 120 × 1 = 120, which is the term standing in place of the question mark.
Factor 1.5: 120 × 1.5 = 180, and 180 is exactly the given fourth term, so the step of 0.5 is confirmed rather than assumed.
Factor 2: 180 × 2 = 360, the given fifth term.
Factor 2.5: 360 × 2.5 = 900, the given sixth term.
Cross-check — Each listed number can be tested by the two factors it would create on either side of the gap:
Number in the gap | Factor from 120 to it | Factor from it to 180 |
|---|---|---|
120 | 120 ÷ 120 = 1 | 180 ÷ 120 = 1.5 |
130 | 130 ÷ 120 ≈ 1.083 | 180 ÷ 130 ≈ 1.385 |
150 | 150 ÷ 120 = 1.25 | 180 ÷ 150 = 1.2 |
Only 120 keeps the step of 0.5 running through the whole factor list (0.5, 1, 1.5, 2, 2.5). With 130 the two factors differ by about 0.30, and with 150 the second factor is smaller than the first, so in both of those cases the factors stop forming an arithmetic progression. Exactly one of the listed numbers therefore fits, which also settles the “more than one of the above” and “none of the above” choices.
Result — The completed series is 240, 120, 120, 180, 360, 900, so the number replacing the question mark is 120.